Wall-crossing on universal compactified Jacobians

计数几何是数学中最古老的领域之一，它的目的是计算具有一定性质的几何对象的个数。例如，我们可能会问有多少条直线穿过平面上的两个给定点。正是欧几里得的第一条公理断言，存在一条唯一的这样的线。另一个例子是计算有多少点同时属于平面上的两条直线。在这里，欧几里得的第五条公理实质上暗示，当且仅当直线不平行时，答案是1。举一个稍微有趣的例子，你可以考虑平面上的抛物线和圆，发现属于这两条线的点的数量可以是0到4之间的任何数字(取决于直线和圆的相对位置)。上面的例子有望说明这样的问题是如何在几何学中基本和普遍存在的，它们让我们得以一窥几何学的早期历史发展。今天，这个领域仍然存在，非常活跃，它使用了来自不同数学领域的技术。在过去的25年里，这一领域的革命性想法来自物理学，特别是源于寻求统一四个基本力的理论，如弦理论。今天计算理论的主要现代方法使用模空间。平面上有多少个二次曲线通过5个普通点？一种可能的方法是考虑将平面二次曲面参数化的5维(射影)空间，并认识到通过点的约束对应于在该空间中切割超平面。通过5个超平面的相交，我们发现计数问题的答案是1。本研究遵循这一范式来探讨代数几何中的一些问题。本文所研究的模空间是射影代数曲线上某一固定次数的线丛的模，约束条件是这样的线丛具有给定数目的线性独立的整体截面(Brill-Noether轨迹)。为了构造这样的模空间，必须引入一个“额外的”参数，而不是由前述几何对象的参数化问题强加的先验参数，称为稳定性。该参数是一个连续参数，但实际上只有当该参数穿过其所在空间中的一些超平面(称为墙)时，模数空间才会发生变化。我们的观点是，当我们考虑所有稳定性参数时，几何图形应该简化，而不是只考虑一个参数。例如，通常有一个“容易”的参数，它的给定约束及其几何性质可以很容易地理解，而有一个“有趣的”参数受到了几位数学家的大量关注。我们方法的新奇之处在于，通过首先解决与“简单”参数相同的问题，找到与“有趣”参数相对应的模空间的结果，然后研究当穿过墙时，模空间如何随稳定性参数变化。不同的模数空间应该通过翻转相互关联(进入墙应该对应于收缩)。

Enumerative geometry is one of the most ancient fields of mathematics, and it aims at counting the number of geometric objects having a certain property. For example, we may ask how many straight lines pass through two given points in the plane. It is Euclid's very first axiom that asserts that there is a unique such line. Another example is to count how many points belong simultaneously to two lines in the plane. Here Euclid's fifth axiom essentially implies that the answer is one if and only if the lines are not parallel. For a slightly more interesting example, one could consider a parabola and a circle in the plane, and see that the number of points belonging to both could be any number between 0 and 4 (depending on the relative position of the line and the circle.The examples above hopefully demonstrate how such questions can be basic and pervasive in geometry, and they give a glimpse onto geometry's early historical developments. Today the field is still existing and very active, and it employs techniques coming from different fields of mathematics. In the last 25 years, revolutionary ideas in the field have arrived from physics, in particular from theories originating from the quest of unifying the four fundamental forces, like string theory.The main modern approach to counting theories today uses moduli spaces. How many quadrics pass through 5 general points in the plane? A possible approach is to consider the 5-dimensional (projective) space that parametrizes plane quadrics, and to realize that the constraint of passing through a point corresponds to cutting a hyperplane in such space. By intersecting the 5 hyperplanes, we find out that the answer to the counting question is 1. The proposed research follows this paradigm to approach some questions in algebraic geometry. The moduli spaces studied in this proposal are moduli of line bundles of some fixed degree over projective algebraic curves, and the constraints are given (for example) by imposing that such line bundles have a given number of linearly independent global sections (Brill-Noether loci). In order to construct such moduli spaces one has to introduce an "extra" parameter, not a-priori imposed by the problem of parameterizing the aforementioned geometric objects, called stability. This parameter is a continuous parameter, but the moduli space actually varies only when the parameter crosses some hyperplanes (called walls) in the space where it lives. Our point of view is that the geometric picture should simplify when one considers all stability parameters, rather than only one. For example, there is usually one "easy" parameter, for which the given constraints and their geometric nature can be easily understood and there is one "interesting" parameter that has received lots of attention from several mathematicians. The novelty of our approach consists in finding results for the moduli space corresponding to the "interesting" parameter by first solving the same problem for the "easy" parameter, and then investigating how the moduli spaces vary with the stability parameter when a wall is crossed. The different moduli spaces should be related to each other by flips (and going into a wall should correspond to a contraction).